I don’t want to repeat the Yoneda lemma, but one of the important corollary should be emphasized. Roughly speaking, given any object in the category , is determined by its morphisms up to isomorphism. That is, the natural isomorphism between hom-sets corresponds to the isomorphism between objects, i.e. . How do we prove it? Well, let’s see a simpler analogy in partial orders.

Lemma. if and only if .Proof. If , then and which imply and . Hence . The converse is quite trivial.

First, we can prove it by brute force method.

Lemma. if and only if .Proof. Consider the following diagrams,

by replacing with , we will see that and we will have the other isomorphism by replacing with and reversing the arrows.

However, the diagram above is exactly used in Yoneda lemma, and of course we can use Yoneda lemma to prove it.

First recall that Yoneda lemma states that where is a Set-valued functor and is an object in a locally small category . Let be the hom-set functor . It gives the isomorphism between . Hence, it indeed tells you that is fully faithful. Therefore, we have the second proof.

Proof2. Given that is fully faithful, thus the natural isomorphisms actually correspond to the isomorphism and vice verse. That’s it!